How would you describe John Nash during the first third of the film A Beautiful Mind? What is his personality like? How did he view himself? How did he interact with others?

John Nash at the beginning of the film A Beautiful Mind is incredibly brilliant, successful, and arrogant. He is a prestigious graduate student at Princeton with a very promising career ahead of him. His intellect is unparalleled, and he quickly creates some profound new mathematical theories, particularly the "Nash Equilibrium," which is a novel approach to Game Theory. He is very socially inept, however, and has difficulty connecting with others, in part because of his awkwardness and in part due to his ego.
Because of his intellect, he insists on publishing an original theory instead of rehashing something that has already been examined, which leads him to publish his theory of the "Nash Equilibrium." He begins, purportedly, to work for the American government, deciphering codes left by Soviet spies. As he does this, he grows more reclusive and paranoid, although he does meet and marry Alicia Larde. Soon after their marriage, it is revealed that Nash has paranoid schizophrenia and that he has hallucinated large amounts of his activities, including his involvement with the US government.

John Nash is a brilliant mathematician who ends up developing a mental disorder that derails his studies later in life. The beginning third of the film outlines his time in school when he is working towards his Ph.D. Because of his undeniable brilliance and the prodigious nature of his work, he turns out to be extremely arrogant.
He is clearly an incredible individual with advanced mental capabilities. Unfortunately, however, this is to his detriment, because he is very self-centered and proud in the beginning of the story, showing disregard for others: he is clearly egotistical. This eventually changes when he falls in love and meets others who can challenge him intellectually, but there is not a definitive turning point until his mental illness starts to derail his life.

The first third of A Beautiful Mind encompasses John Nash's time at Princeton as a PhD student and his early work at the MIT and the Defense Department. Nash at Princeton is portrayed as extraordinarily drive—he is obsessed with coming up with what he refers to as an "original idea" for his dissertation—and very socially awkward. He is also undeniably brilliant, as his "original idea" demonstrates, and he knows it—although he displays a great deal of self-doubt. At MIT, Nash is arrogant and utterly self-confident. He treats the classes he is supposed to teach with contempt and focuses on his classified work for the US military. He has little interaction with other people besides his immediate colleagues, who defer to him on most things. He begins to have what the viewer learns are serious delusions about working as a codebreaker for the Department of Defense. It is here that, despite his obvious awkwardness and arrogance, he meets Alicia, who, in the movie, would become the love of his life. Later in the film, we discover that much about Nash's life in the first third of the movie existed only in his mind, which is of course crucial to understanding his character.

At the beginning of the film, Nash's character is difficult yet still sympathetic. He has a prickly personality. He is well aware of his gift, and at the same time, he is frustrated by the challenges and burdens it poses. He is aware that he is like a magnificent swan running around with common pigeons, in terms of his intellect. This leads to the disagreeable side of his personality coming out. He apparently views himself as above most others, whom he refers to as "ordinary mortals."
He often lives in his own head and thereby shuts out a lot of ordinary human interaction. In fairness, shutting out ordinary human interaction is not his primary intention—his primary intention is using his gifts to think, and he has the gears turning all the time. Because of this, he is perceived as odd and standoffish.
He is quirky and hard to reach, but he eventually finds ways to open up to others.
My suggestion would be to use at least two or three specific scenes from the movie to underscore your points about Nash's character. Good luck!